Rig a pre-bent curve to smoothly unbend
The problem is that you have too many control points on the curve, which makes it hard to control.

Would be best if you could fit a bezier or bspline curve to the shape of your curve with less control points.

You can control curve(s) with vertex hooks modifiers pointing to bones. Add an armature and make a bone in every curve's control vertex location. Select bone in posemode, shift-select curve, enter edit mode, select control vertex at bone's location (or multiple points if there are many in line), Ctrl+H+Make Hook to Pose Bone.

Then pose the bones so they are straight. You can parent them into a chain and position them by rotating the parents, so the curve points move naturally in an arch.

The more control points the more tedious it is. This process can be however scripted easily.